Proof.   Let \(U=(u_1,...,u_{2s})\)  be an element of
\(\mathcal {S}_\Phi \)  with a \(2s\) -point \(a_U\) , then
there exist \(2s-1\)  or \(2s\)  arcs connecting \(a_U\) 
to any of the isolated zeros of the boundary datum
(note that there are \(2s-1\)  arcs if and only if
\(a_U\in \partial D\) ). Then the function
\(\psi _a=\sum _{j=1}^{2s}(-1)^ju_j\)  is harmonic in
\(D\setminus \lbrace a_U\rbrace \)  (see [1]}),
moreover \(\psi _a\)  is bounded so, by Schwarz's removable
singularity principle (see [2]})
\(\psi _a\)  is harmonic in \(D\)  and, by construction,
\(U=|\psi _a|\)  in \(D\)  and \(U=|\Phi ^a|=\Phi \)  on
\(\partial D\) .
